Pair State Analysis of the Hubbard Hamiltonian in One-Dimension

Recently, there has been renewed interest in using a variational determination of the two-particle reduced density matrix $^2D$ to find the ground state energy of an $N$-electron system. This interest can be partly attributed to progress in solving this constrained optimization problem using semidefinite programming algorithms (SDPA).\footnote{Nakata, Nakatsuji, and co-workers, J. Chem. Phys. {\bf{114}}, 8282 (2001), Hammond and Mazziotti, Phys. Rev. A {\bf{73}}, 062505 (2006).} We use the one-dimensional Hubbard model for comparing several variations of the SPDA appproach with the exact results, considering either even or odd numbers $N$ of electrons, either periodic or fixed boundary conditions, and various values of the Coulomb energy parameter $U/t$. It is convenient to use the two-electron eigenstates of the pair Hubbard Hamiltonian as a basis for representing the $^2D$ matrix using the normalization ${\rm{Tr}}(^2D)= N(N-1)/2$. For example, for $N=4$ at half filling, using all of the two-particle constraints along with the appropriate physical constraints, we find the SDPA ground state energies to differ from the exact ones by less than $4\times10^{-4}t$. In addition, the diagonal elements of $^2D$ generally differ from the exact ones on the order of $10^{-2}$.